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# Expected number of moves to reach the end of a board | Matrix Exponentiation

• Last Updated : 09 Jun, 2021

Given a linear board of length N numbered from 1 to N, the task is to find the expected number of moves required to reach the Nth cell of the board, if we start at cell numbered 1 and at each step we roll a cubical dice to decide the next move. Also, we cannot go outside the bounds of the board. Note that the expected number of moves can be fractional.

Examples:

Input: N = 8
Output:
p1 = (1 / 6) | 1-step -> 6 moves expected to reach the end
p2 = (1 / 6) | 2-steps -> 6 moves expected to reach the end
p3 = (1 / 6) | 3-steps -> 6 moves expected to reach the end
p4 = (1 / 6) | 4-steps -> 6 moves expected to reach the end
p5 = (1 / 6) | 5-steps -> 6 moves expected to reach the end
p6 = (1 / 6) | 6-steps -> 6 moves expected to reach the end
If we are 7 steps away, then we can end up at 1, 2, 3, 4, 5, 6 steps
away with equal probability i.e. (1 / 6).
Look at the above simulation to understand better.
dp[N – 1] = dp[7]
= 1 + (dp[1] + dp[2] + dp[3] + dp[4] + dp[5] + dp[6]) / 6
= 1 + 6 = 7

Input: N = 10
Output: 7.36111

Approach: An approach based on dynamic programming has already been discussed in an earlier post. In this article, a more optimized method to solve this problem will be discussed. The idea is using a technique called Matrix-Exponentiation
Let’s define An as the expected number of moves to reach the end of a board of length N + 1

The recurrence relation will be:

An = 1 + (An-1 + An-2 + An-3 + An-4 + An-5 + An-6) / 6

Now, the recurrence relation needs to be converted in a suitable format.

An = 1 + (An-1 + An-2 + An-3 + An-4 + An-5 + An-6) / 6 (equation 1)
An-1 = 1 + (An-2 + An-3 + An-4 + An-5 + An-6 + An-7) / 6 (equation 2)
Substracting 1 with 2, we get An = 7 * (An-1) / 6 – (An-7) / 6

Matrix exponentiation technique can be applied here on the above recurrence relation.
Base will be {6, 6, 6, 6, 6, 6, 0} corresponding to {A6, A5, A4…, A0}
Multiplier will be

{7/6, 1, 0, 0, 0, 0, 0},
{0, 0, 1, 0, 0, 0, 0},
{0, 0, 0, 1, 0, 0, 0},
{0, 0, 0, 0, 1, 0, 0},
{0, 0, 0, 0, 0, 1, 0},
{0, 0, 0, 0, 0, 0, 1},
{-1/6, 0, 0, 0, 0, 0, 0}

To find the value:

• Find mul(N – 7)
• Find base * mul(N – 7).
• The first value of the 1 * 7 matrix will be the required answer.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``#define maxSize 50``using` `namespace` `std;` `// Function to multiply two 7 * 7 matrix``vector > matrix_product(``    ``vector > a,``    ``vector > b)``{``    ``vector > c(7);``    ``for` `(``int` `i = 0; i < 7; i++)``        ``c[i].resize(7, 0);` `    ``for` `(``int` `i = 0; i < 7; i++)``        ``for` `(``int` `j = 0; j < 7; j++)``            ``for` `(``int` `k = 0; k < 7; k++)``                ``c[i][j] += a[i][k] * b[k][j];``    ``return` `c;``}` `// Function to perform matrix exponentiation``vector > mul_expo(vector > mul, ``int` `p)``{` `    ``// 7 * 7 identity matrix``    ``vector > s = { { 1, 0, 0, 0, 0, 0, 0 },``                                  ``{ 0, 1, 0, 0, 0, 0, 0 },``                                  ``{ 0, 0, 1, 0, 0, 0, 0 },``                                  ``{ 0, 0, 0, 1, 0, 0, 0 },``                                  ``{ 0, 0, 0, 0, 1, 0, 0 },``                                  ``{ 0, 0, 0, 0, 0, 1, 0 },``                                  ``{ 0, 0, 0, 0, 0, 0, 1 } };` `    ``// Loop to find the power``    ``while` `(p != 1) {``        ``if` `(p % 2 == 1)``            ``s = matrix_product(s, mul);``        ``mul = matrix_product(mul, mul);``        ``p /= 2;``    ``}` `    ``return` `matrix_product(mul, s);``}` `// Function to return the required count``double` `expectedSteps(``int` `x)``{` `    ``// Base cases``    ``if` `(x == 0)``        ``return` `0;``    ``if` `(x <= 6)``        ``return` `6;` `    ``// Multiplier matrix``    ``vector > mul = { { (``double``)7 / 6, 1, 0, 0, 0, 0, 0 },``                                    ``{ 0, 0, 1, 0, 0, 0, 0 },``                                    ``{ 0, 0, 0, 1, 0, 0, 0 },``                                    ``{ 0, 0, 0, 0, 1, 0, 0 },``                                    ``{ 0, 0, 0, 0, 0, 1, 0 },``                                    ``{ 0, 0, 0, 0, 0, 0, 1 },``                                    ``{ (``double``)-1 / 6, 0, 0, 0, 0, 0, 0 } };` `    ``// Finding the required multiplier``    ``// i.e mul^(X-6)``    ``mul = mul_expo(mul, x - 6);` `    ``// Final answer``    ``return` `(mul[0][0]``            ``+ mul[1][0]``            ``+ mul[2][0]``            ``+ mul[3][0]``            ``+ mul[4][0]``            ``+ mul[5][0])``           ``* 6;``}` `// Driver code``int` `main()``{``    ``int` `n = 10;` `    ``cout << expectedSteps(n - 1);` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``class` `GFG``{``static` `final` `int` `maxSize = ``50``;` `// Function to multiply two 7 * 7 matrix``static` `double` `[][] matrix_product(``double` `[][] a,``                                  ``double` `[][] b)``{``    ``double` `[][] c = ``new` `double``[``7``][``7``];` `    ``for` `(``int` `i = ``0``; i < ``7``; i++)``        ``for` `(``int` `j = ``0``; j < ``7``; j++)``            ``for` `(``int` `k = ``0``; k < ``7``; k++)``                ``c[i][j] += a[i][k] * b[k][j];``    ``return` `c;``}` `// Function to perform matrix exponentiation``static` `double` `[][] mul_expo(``double` `[][] mul, ``int` `p)``{` `    ``// 7 * 7 identity matrix``    ``double` `[][] s = {{ ``1``, ``0``, ``0``, ``0``, ``0``, ``0``, ``0` `},``                     ``{ ``0``, ``1``, ``0``, ``0``, ``0``, ``0``, ``0` `},``                     ``{ ``0``, ``0``, ``1``, ``0``, ``0``, ``0``, ``0` `},``                     ``{ ``0``, ``0``, ``0``, ``1``, ``0``, ``0``, ``0` `},``                     ``{ ``0``, ``0``, ``0``, ``0``, ``1``, ``0``, ``0` `},``                     ``{ ``0``, ``0``, ``0``, ``0``, ``0``, ``1``, ``0` `},``                     ``{ ``0``, ``0``, ``0``, ``0``, ``0``, ``0``, ``1` `}};` `    ``// Loop to find the power``    ``while` `(p != ``1``)``    ``{``        ``if` `(p % ``2` `== ``1``)``            ``s = matrix_product(s, mul);``        ``mul = matrix_product(mul, mul);``        ``p /= ``2``;``    ``}``    ``return` `matrix_product(mul, s);``}` `// Function to return the required count``static` `double` `expectedSteps(``int` `x)``{` `    ``// Base cases``    ``if` `(x == ``0``)``        ``return` `0``;``    ``if` `(x <= ``6``)``        ``return` `6``;` `    ``// Multiplier matrix``    ``double` `[][]mul = { { (``double``)``7` `/ ``6``, ``1``, ``0``, ``0``, ``0``, ``0``, ``0` `},``                                   ``{ ``0``, ``0``, ``1``, ``0``, ``0``, ``0``, ``0` `},``                                   ``{ ``0``, ``0``, ``0``, ``1``, ``0``, ``0``, ``0` `},``                                   ``{ ``0``, ``0``, ``0``, ``0``, ``1``, ``0``, ``0` `},``                                   ``{ ``0``, ``0``, ``0``, ``0``, ``0``, ``1``, ``0` `},``                                   ``{ ``0``, ``0``, ``0``, ``0``, ``0``, ``0``, ``1` `},``                                   ``{(``double``) - ``1` `/ ``6``, ``0``, ``0``,``                                                ``0``, ``0``, ``0``, ``0` `}};` `    ``// Finding the required multiplier``    ``// i.e mul^(X-6)``    ``mul = mul_expo(mul, x - ``6``);` `    ``// Final answer``    ``return` `(mul[``0``][``0``] + mul[``1``][``0``] + mul[``2``][``0``] +``            ``mul[``3``][``0``] + mul[``4``][``0``] + mul[``5``][``0``]) * ``6``;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `n = ``10``;` `    ``System.out.printf(``"%.5f"``,expectedSteps(n - ``1``));``}``}` `// This code is contributed by PrinciRaj1992`

## Python3

 `# Python3 implementation of the approach``import` `numpy as np` `maxSize ``=` `50` `# Function to multiply two 7 * 7 matrix``def` `matrix_product(a, b) :``    ``c ``=` `np.zeros((``7``, ``7``));``    ` `    ``for` `i ``in` `range``(``7``) :``        ``for` `j ``in` `range``(``7``) :``            ``for` `k ``in` `range``(``7``) :``                ``c[i][j] ``+``=` `a[i][k] ``*` `b[k][j];``    ``return` `c` `# Function to perform matrix exponentiation``def` `mul_expo(mul, p) :`  `    ``# 7 * 7 identity matrix``    ``s ``=` `[ [ ``1``, ``0``, ``0``, ``0``, ``0``, ``0``, ``0` `],``          ``[ ``0``, ``1``, ``0``, ``0``, ``0``, ``0``, ``0` `],``          ``[ ``0``, ``0``, ``1``, ``0``, ``0``, ``0``, ``0` `],``          ``[ ``0``, ``0``, ``0``, ``1``, ``0``, ``0``, ``0` `],``          ``[ ``0``, ``0``, ``0``, ``0``, ``1``, ``0``, ``0` `],``          ``[ ``0``, ``0``, ``0``, ``0``, ``0``, ``1``, ``0` `],``          ``[ ``0``, ``0``, ``0``, ``0``, ``0``, ``0``, ``1` `] ];` `    ``# Loop to find the power``    ``while` `(p !``=` `1``) :``        ``if` `(p ``%` `2` `=``=` `1``) :``            ``s ``=` `matrix_product(s, mul);``            ` `        ``mul ``=` `matrix_product(mul, mul);``        ``p ``/``/``=` `2``;` `    ``return` `matrix_product(mul, s);` `# Function to return the required count``def` `expectedSteps(x) :` `    ``# Base cases``    ``if` `(x ``=``=` `0``) :``        ``return` `0``;``        ` `    ``if` `(x <``=` `6``) :``        ``return` `6``;` `    ``# Multiplier matrix``    ``mul ``=` `[ [ ``7` `/` `6``, ``1``, ``0``, ``0``, ``0``, ``0``, ``0` `],``            ``[ ``0``, ``0``, ``1``, ``0``, ``0``, ``0``, ``0` `],``            ``[ ``0``, ``0``, ``0``, ``1``, ``0``, ``0``, ``0` `],``            ``[ ``0``, ``0``, ``0``, ``0``, ``1``, ``0``, ``0` `],``            ``[ ``0``, ``0``, ``0``, ``0``, ``0``, ``1``, ``0` `],``            ``[ ``0``, ``0``, ``0``, ``0``, ``0``, ``0``, ``1` `],``            ``[ ``-``1` `/` `6``, ``0``, ``0``, ``0``, ``0``, ``0``, ``0` `] ];` `    ``# Finding the required multiplier``    ``# i.e mul^(X-6)``    ``mul ``=` `mul_expo(mul, x ``-` `6``);` `    ``# Final answer``    ``return` `(mul[``0``][``0``] ``+` `mul[``1``][``0``] ``+` `mul[``2``][``0``] ``+``            ``mul[``3``][``0``] ``+` `mul[``4``][``0``] ``+` `mul[``5``][``0``]) ``*` `6``;` `# Driver code``if` `__name__ ``=``=` `"__main__"` `:` `    ``n ``=` `10``;` `    ``print``(expectedSteps(n ``-` `1``));``    ` `# This code is contributed by AnkitRai01`

## C#

 `// C# implementation of the approach``using` `System;` `class` `GFG``{``static` `readonly` `int` `maxSize = 50;` `// Function to multiply two 7 * 7 matrix``static` `double` `[,] matrix_product(``double` `[,] a,``                                 ``double` `[,] b)``{``    ``double` `[,] c = ``new` `double``[7, 7];` `    ``for` `(``int` `i = 0; i < 7; i++)``        ``for` `(``int` `j = 0; j < 7; j++)``            ``for` `(``int` `k = 0; k < 7; k++)``                ``c[i, j] += a[i, k] * b[k, j];``    ``return` `c;``}` `// Function to perform matrix exponentiation``static` `double` `[,] mul_expo(``double` `[,] mul, ``int` `p)``{` `    ``// 7 * 7 identity matrix``    ``double` `[,] s = {{ 1, 0, 0, 0, 0, 0, 0 },``                    ``{ 0, 1, 0, 0, 0, 0, 0 },``                    ``{ 0, 0, 1, 0, 0, 0, 0 },``                    ``{ 0, 0, 0, 1, 0, 0, 0 },``                    ``{ 0, 0, 0, 0, 1, 0, 0 },``                    ``{ 0, 0, 0, 0, 0, 1, 0 },``                    ``{ 0, 0, 0, 0, 0, 0, 1 }};` `    ``// Loop to find the power``    ``while` `(p != 1)``    ``{``        ``if` `(p % 2 == 1)``            ``s = matrix_product(s, mul);``        ``mul = matrix_product(mul, mul);``        ``p /= 2;``    ``}``    ``return` `matrix_product(mul, s);``}` `// Function to return the required count``static` `double` `expectedSteps(``int` `x)``{` `    ``// Base cases``    ``if` `(x == 0)``        ``return` `0;``    ``if` `(x <= 6)``        ``return` `6;` `    ``// Multiplier matrix``    ``double` `[,]mul = {{(``double``)7 / 6, 1, 0, 0, 0, 0, 0 },``                                ``{ 0, 0, 1, 0, 0, 0, 0 },``                                ``{ 0, 0, 0, 1, 0, 0, 0 },``                                ``{ 0, 0, 0, 0, 1, 0, 0 },``                                ``{ 0, 0, 0, 0, 0, 1, 0 },``                                ``{ 0, 0, 0, 0, 0, 0, 1 },``                                ``{(``double``) - 1 / 6, 0, 0,``                                                ``0, 0, 0, 0 }};` `    ``// Finding the required multiplier``    ``// i.e mul^(X-6)``    ``mul = mul_expo(mul, x - 6);` `    ``// Final answer``    ``return` `(mul[0, 0] + mul[1, 0] + mul[2, 0] +``            ``mul[3, 0] + mul[4, 0] + mul[5, 0]) * 6;``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``int` `n = 10;` `    ``Console.Write(``"{0:f5}"``, expectedSteps(n - 1));``}``}` `// This code is contributed by Rajput-Ji`

## Javascript

 ``
Output:
`7.36111`

Time Complexity of the above approach will be O(343 * log(N)) or simply O(log(N)).

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